×

zbMATH — the first resource for mathematics

Analytical and numerical inversion of the Laplace-Carson transform by a differential method. (English) Zbl 0998.65127
Summary: A differential method is presented for recovering a function from its Laplace-Carson transform \(p\widehat f(p)\) given as continuous or discrete data on a finite interval. The introduction of the variable \(u=1/p\) converts this transform into a Mellin convolution, with a transformed kernel involving the gamma function \(\Gamma\). The truncation of the infinite product representation of \(1/\Gamma\) leads to an approximate differential expression for the solution. The algorithm is applied to selected analytical and numerical test problems; discrete and noisy data are differentiated with the aid of Tikhonov’s regularization. For the inversion of a Laplace transform, the present formula is proved to be equivalent to the Post-Widder expression.

MSC:
65R10 Numerical methods for integral transforms
44A10 Laplace transform
65R32 Numerical methods for inverse problems for integral equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Sveshnikov, A.G.; Tikhonov, A.N., The theory of functions of a complex variable, (1978), Mir Publishers Moscow, p. 224 · Zbl 0212.42101
[2] Makarov, A.M.; Christensen, R.M., Application of the laplace – carson method of integral transformation to the theory of unsteady visco-plastic flows, J. engrg. phys. thermophys., Theory of viscoelasticity, 19, 94-99, (1982), Academic Press New York
[3] Craig, I.J.D.; Brown, J.C., Inverse problems in astronomy, (1986), Hilger Bristol
[4] Donolato, C., Reconstruction of the charge collection probability in a solar cell from internal quantum efficiency measurements, J. appl. phys., 89, 5687-5695, (2001)
[5] Bertero, M.; Brianzi, P.; Pike, E.R., On the recovery and resolution of exponential relaxation rates from experimental data. III. the effect of sampling and truncation of data on the Laplace transform inversion, Proc. roy. soc. London A, 398, 23-44, (1985) · Zbl 0611.65089
[6] Brianzi, P.; Fortini, M., On the regularized inversion of the Laplace transform, Inverse probl., 7, 1355-1368, (1991)
[7] Cunha, C.; Viloche, F., The Laguerre functions in the inversion of the Laplace transform, Inverse probl., 79, 57-68, (1993) · Zbl 0766.65118
[8] Iqbal, M., On a numerical technique regarding the inversion of the Laplace transform, J. comput. appl. math., Comput. phys. commun., 88, 43-50, (1995), On comparison of spline regularization with exponential sampling method for Laplace transform inversion · Zbl 0888.65136
[9] Istratov, A.A.; Vyvenko, O.F., Exponential analysis in physical phenomena, Rev. sci. instrum., 70, 1233-1257, (1999)
[10] Bellman, R.; Kalaba, R.E.; Lockett, J.A., Numerical inversion of the Laplace transform, (1966), American Elsevier New York · Zbl 0147.14003
[11] Davies, B.; Martin, B., Numerical inversion of the Laplace transform: a survey and comparison of methods, J. comput. phys., 33, 1-32, (1979) · Zbl 0416.65077
[12] Sjøntoft, E., A straightforward deconvolution method for use in small computers, Nucl. instrum. methods, 163, 519-522, (1979)
[13] Vasudeva Murthy, A.S., A note on the differential inversion method of hohlfield et al., SIAM J. appl. math., 55, 719-722, (1995) · Zbl 0838.45003
[14] Tikhonov, A.N.; Arsenin, V.Y., Solutions to ill-posed problems, (1977), Wiley New York · Zbl 0354.65028
[15] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), McGraw-Hill New York, pp. 484-485
[16] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover New York · Zbl 0515.33001
[17] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, (1980), Academic London · Zbl 0521.33001
[18] Post, E.L., Generalized differentiation, Trans. amer. math. soc., 32, 723-781, (1930) · JFM 56.0349.01
[19] Widder, D.V., The Laplace transform, (1946), University Press Princeton, Chapter VII · Zbl 0060.24801
[20] Rall, L.B.; Corliss, G.F., An introduction to automatic differentiation, (), 1-18 · Zbl 0940.65019
[21] Griewank, A., Evaluating derivatives: principles and techniques of algorithmic differentiation, (2000), SIAM Philadelphia, PA · Zbl 0958.65028
[22] Jagerman, D.L., An inversion technique for the Laplace transform with application to approximation, Bell syst. tech. J., 57, 669-710, (1978) · Zbl 0395.44003
[23] Frolov, G.A.; Kitaev, M.Y., Improvement of the accuracy in numerical methods for inverting Laplace transforms based on the post – widder formula, Comput. math. appl., 36, 23-34, (1998) · Zbl 0932.65132
[24] Carnahan, B.; Luther, H.A.; Wilkes, J.O., Applied numerical methods, (1969), Wiley New York, p. 78 · Zbl 0195.44701
[25] Dahlquist, G.; Björck, Å., Numerical methods, (1974), Prentice-Hall Englewood Cliffs, NJ
[26] Glasko, V.G., Inverse problems of mathematical physics, (1988), American Institute of Physics New York, p. 65 · Zbl 0685.35001
[27] Feller, W., An introduction to probability theory and its applications, II, (1971), Wiley New York, p. 233 · Zbl 0219.60003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.